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The wide adoption of DNNs has given birth to unrelenting computing requirements, forcing datacenter operators to adopt domain-specific accelerators to train them. These accelerators typically employ densely packed full-precision floating-point arithmetic to maximize performance per area. Ongoing research efforts seek to further increase that performance density by replacing floating-point with fixed-point arithmetic. However, a significant roadblock for these attempts has been fixed point's narrow dynamic range, which is insufficient for DNN training convergence. We identify block floating point (BFP) as a promising alternative representation since it exhibits wide dynamic range and enables the majority of DNN operations to be performed with fixed-point logic. Unfortunately, BFP alone introduces several limitations that preclude its direct applicability. In this work, we introduce HBFP, a hybrid BFP-FP approach, which performs all dot products in BFP and other operations in floating point. HBFP delivers the best of both worlds: the high accuracy of floating point at the superior hardware density of fixed point. For a wide variety of models, we show that HBFP matches floating point's accuracy while enabling hardware implementations that deliver up to 8.5 x higher throughput.